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## Technical Papers Authored or Co-authored By Dr. Robert H. Badgley

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1. Badgley, R. H. and Booker, J. F. , **"Turborotor
Instability - Effect of Initial Transients on Plane Motion, " ** ASME Paper No.
68-Lub-7, presented at the ASME-ASLE Lubrication Conference, Atlantic City,
N. J. , October 8-10, 1968, and published in Trans. ASME, Journal of Lubrication
Technology, Vol. 91, Series F, No. 4, October 1969, p. 625-633.

The rigid-body dynamics of rotors supported in plain, cylindrical, cavitated, fluid-film journal bearings are investigated numerically by Runge-Kutta extrapolation techniques. Expressions for journal force due to the fluid-film are developed for the short-bearing (Ocvirk), long-bearing (Sommerfeld), and finite-length-bearing (Warner) approximate solutions to the Reynolds equation. Stability of Plane motion is investigated for each solution under the assumption of light initial impact. The long-bearing solution appears to be most conservative (that is, it predicts the onset of instability at lower angular velocity ratios than the other solutions) for static eccentricity ratios between a and 0. 5, while the finite-bearing solution, with bearing length-to-diameter ratio L/D equal to 1, appears most conservative at higher static eccentricity ratios. Variations in L/D between 0. 5 and 2. 0 appear not to affect journal path shapes appreciably. Variations in initial journal center velocity are found to be important, at least with the short-bearing solution: large initial velocities are observed to produce instability for certain parameter combinations which are stable under small initial position or velocity disturbances. In all cases investigated, instability is not observed above static eccentricity ratios of 0. 83.

2. Badgley, R. H. And Booker, J. F. , **"Rigid-Body Rotor Dynamics:
Dynamic Unbalance and Lubricant Temperature Changes, " ** ASME Paper
No. 69-Lub-14, presented at the ASME-ASLE Lubrication Conference, Houston,
Texas, October 14-16, 1969, and published in Trans. ASME, Journal of Lubrication
Technology, Vol. 92, Series F, No. 3, July 1970, p. 415-424.

The case of a symmetric rotor supported on two identical, rigidly mounted, self-aligning, finite-length (L/D =1) fluid-film journal bearings is considered. Rotor position is described by two translation coordinates in a plane perpendicular to the bearing line of centers, and by three Euler angles. Introduction of various amounts of dynamic unbalance via the inertia tensor off-diagonal terms (products of inertia) allows determination of angular velocity and static eccentricity ratio combinations leading to bearing "failure" defined for arbitrary maximum allowable eccentricity ratios. Instability hysteresis, defined here as the persistence, during rotor deceleration, of instability to speeds below which it first appeared, is considered by means of the above model. Equations and methods developed for the unbalance investigation are adapted to a variable-speed analysis. With both constant and variable mean bearing temperatures, variable-speed simulations terminating at constant speed are observed to be stable when the terminating point is below the instability threshold curve on the angular velocity-static eccentricity ratio parameter plane and unstable when above. The slope of the threshold curve and the shape of the equilibrium-condition path on the parameter plane (single-line path for constant temperature, closed curve for variable temperature) apparently combine to produce hysteresis in the variable temperature case and none at constant temperature.

3. Badgley, R. H. , **"Mechanical Aspects of Gear-Induced
Noise in Complete Power Train Systems, " ** ASME Paper No. 70-WA/DGP-1, presented
at the ASME Winter Annual Meeting, November 29 - December 3, 1970, New York,
N. Y.

The author presents an in-depth systems approach to the problem of interior helicopter noise, showing that step-by-step consideration of the flow of vibration energy within the drive train yields both qualitative understanding of the problem and technology required to alleviate it. Since precise methods for computing interior noise from design and operating data, are more economical than cut and try techniques for evaluating proposed noise-reduction design changes, the development of precise methods is stressed in the study. In-flight vibration and noise measurements, gathered to form a data bank, are used to assist in understanding the vibration energy flow and assessing adequacy and accuracy of the analytical procedures (comparison of calculated and measured noise spectra). The author uses laboratory experiments on parts of the system to study the behavior of critical components. The overall study, directed specifically at UH-1D and CH-47 helicopters, is coordinated and controlled by a comprehensive plan and schedule designed to ensure inclusion of all aspects of the problem.

4. Tessarzik, J. M. , Badgley, R. H. And Anderson, W. J. , **"Flexible
Rotor Balancing by the Exact Point-Speed Influence Coefficient Method, "
** ASME Paper No. 71-Vibr-91, presented at the ASME Vibrations Conference,
Toronto, Canada, September 8-10, 1971, and published in Trans. ASME, Journal
of Engineering for Industry, Vol. 94, Series B, No. 1, February 1972,
p. 148-158.

A test program was conducted to confirm experimentally the validity of the exact point-speed influence coefficient method for balancing rotating machinery, and to assess the practical aspects of applying the method to flexible rotors. Testing was performed with a machine having a 41-in, long, 126-lb rotor. The rotor was operated over a speed range encompassing three rotor-bearing system critical speeds: two" rigid-body" criticals and one flexural critical. Rotor damping at the flexural critical was very low due to the journal bearings being located at the nodal points of the shaft. The balancing method was evaluated for three different conditions of initial rotor unbalance. The method was found to be effective and practical. Safe passage through all the critical speeds was obtained after a reasonable number of balancing runs. Success of the balancing method was, in large part, due to the accuracy of the instrumentation system used to obtain phase-angle measurements during the balancing procedure.

5. Badgley, R. H. , **"Gearbox Dynamics - The Key
to Understanding and Reducing Acoustic-Frequency Energy in Geared Power Trains, " **
presented at the Meeting of the Aerospace Gearing Committee of the American
Gear Manufacturers Association, Cleveland, Ohio, January 17-18, 1972.

The problem of interior noise in large helicopters is most effectively attacked by a systematic, step-by-step study of the flow of high-frequency vibration energy within the rotor drive train. Such a study, which begins with an examination of the mechanism by which high-frequency disturbances are generated at the gear meshes, includes in due course a study of the responses to these disturbances (both torsional and lateral) of the drive train shafts and shaft support elements. Comparison of predicted vibration amplitudes and dynamic stresses with carefully-measured values can be expected to yield both a qualitative understanding of the noise problem and also solution techniques which can be applied to other designs. Such a comparison is in fact underway for the Boeing-Vertol CH-47 helicopter forward rotor gearbox. This paper presents the results of calculations of the vibration response of the CH-47 forward rotor gearbox spiral-bevel gear shafts to spiral-bevel mesh-induced disturbances. These calculations have shown logical reasons why noise is generated by the gearbox at the spiral-bevel mesh frequency.

6. Badgley, R. H. , **"Reduction of Noise and Acoustic-Frequency
Vibrations in Aircraft Transmissions, " ** AHS Paper No. 661, presented at the 28^{th}
Annual National Forum of the American Helicopter Society, Washington, D. C. ,
May 1972.

The most objectionable components of helicopter interior noise are usually generated by either the accessory systems or the rotor drive-train gear boxes. In small and medium-size helicopters, the accessories are often as important as the rotor- drive components with regard to noise, whereas in large helicopters the latter sources are generally more important. The problem of rotor drive-train gearbox noise is most amenable to solutions which involve a systematic, step-by-step study of the flow of high-frequency vibration energy within the drive train. Such studies, which begin with an examination of the mechanism by which mesh-frequency disturbances are generated, must include in due course a ·study of the response (both torsional and lateral)of the drive-train shafts and shaft sup- port elements; This paper presents the results of calculations of the vibration response to spiral- bevel mesh-induced disturbances for the spiral- bevel gearshafts in the Boeing-Vertol CH-47 forward rotor gearbox and the Bell UH-lD main rotor-drive gearbox. The calculations indicate both logical reasons why noise is generated by these gearboxes at the bevel mesh frequencies and also the effects of typical shaft-bearing system design changes which may be useful for noise reduction at those frequencies. Comparison of predicted vibration amplitudes with measured values can be expected to yield both a qualitative understanding of the noise problem and also verified solution techniques which can be applied to other designs.

During the past several years the problem of helicopter internal noise has become of increasing concern to the U. S. Army as the effects of noise on both health and mission performance became better understood. Active support for efforts to under- stand the sources of. noise and to reduce noise levels in operating helicopter aircraft has been provided by the Eustis Directorate of the U. S. Army Air Mobility Research and Development Laboratory (formerly U. S. Army Aviation Materiel Laboratories). These efforts have been directed at the reduction of noise generated by rotor-drive gearboxes, which are major sources of noise in helicopter aircraft. The primary thrust of these efforts has been at the development and verification of analytical proce- dures for dealing directly with the major causes of gearbox noise at their sources:(1)the deviations from uniform gear rotation (referred to herein as "excitations") which are present in even the high- est quality gears under ncrrmal operating conditions, and (2) the amplification of these deviations by the dynamic properties (torsional and lateral)of the drive train components, resulting in loud- speaker-type behavior of the gearbox casing or even the airframe. A considerable measure of success has been achieved to date in obtaining realistic predictions of the levels of gear mesh excitation and the resulting dynamic behavior of the gearbox components.

7. Rieger, N. F. and Badgley, R. H. , **"Flexible Rotor Balancing of
a High-Speed Gas Turbine Engine, " ** SAE Paper No. 720741, presented
at the Society of Automotive Engineers National Combined Farm Construction
& Industrial Machinery and Powerplant Meetings, Milwaukee, Wisconsin,
September, 11-14, 1972.

ABSTRACT: The need for balancing a high-speed rotor in a manner that accounts for its speed-dependent deformations is discussed. The influence coefficient method of flexible rotor balancing is described with reference to the balancing of an advanced gas turbine engine rotor. This engine rotor-bearing system is then studied in detail as an application of flexible rotor bal-ancing, using the influence coefficient method. The relative effectiveness of various combinations of balance speeds and numbers of balance planes is compared...

8. Gu, A. L. , Pan, C. H. T. and Badgley, R. H. , **"Dynamic Stability
of Gimbaled Spiral-Grooved Thrust Bearing, " ** ASME Paper No. 72-Lub-13,
presented at the ASME-ASLE International Lubrication Conference, New York,
N. Y. , October 9-12, 1972, and published in Trans. ASME, Journal of Lubrication
Technology, Vol. 95, Series F, No. 2, April 1973, p. 222-235.

A general, easily implemented technique is developed by which stability maps may be determined for gimbaled, gas-lubricated, spiral-grooved thrust bearings. This technique is based upon the spectral analysis (frequency domain) method in which the neutrally stable states of the stator-gimbal system are determined through solution of the system's characteristic equations. The method has proven effective for conducting low-cost investigations of the sensitivity of system instability thresholds to changes in various system design parameters. Containing data valid for a wide range of gimbal inertias, stability maps are presented for a range of bearing compressibility numbers, for several bearing geometries, and for several values of mechanical damping in order to illustrate both the power of the technique and also its effectiveness. Limited experimental information which is presently available verified the essential features of the corresponding stability maps. The technique has been reduced to a form usable in the design of gas bearings for use in gas turbine engines.

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